Lattice dynamics and inelastic neutron scattering from forsterite,Mg2SiO4: Phonon dispersion relation,density of states and specific heat

Magnesium-rich olivine (Mg_0.9Fe_0.1)₂SiO₄ is considered to be a major constituent of the Earth's upper mantle. Because of its major geophysical importance, the temperature and pressure dependence of its crystal structure, elastic and dielectric constants, long-wavelength phonon modes and specific heat have been measured using a variety of experimental techniques. Theoretical study of lattice dynamics provides a means of analyzing and understanding a host of such experimental data in a unified manner. A detailed study of the lattice dynamics of forsterite, Mg₂SiO₄, has been made using a crystal potential function consisting of Coulombic and short-range terms. Quasiharmonic lattice dynamical calculations based on a rigid molecular-ion model have provided theoretical estimates of elastic constants, long-wavelength modes, phonon dispersion relation for external modes along the three high symmetry directions in the Brillouin zone, total and partial density of states and inelastic neutron scattering cross-sections. The neutron cross-sections were used as guides for the coherent inelastic neutron scattering experiment on a large single crystal using a triple axis spectrometer in the constant Q mode. The observed and predicted phonon dispersion relation show excellent agreement. The inelastically scattered neutron spectra from a powder sample have been analyzed on the basis of a phonon density of states calculated from a rigid-ion model, which includes both external and internal modes. The experimental data from a powder sample show good agreement with the calculated spectra, which include a multiphonon contribution in the incoherent approximation. The computed phonon densities of states are used to calculate the specific heat as a function of temperature using both the rigid molecular-ion and rigid ion models. These results are in very good agreement with the calorimetric measurement of the specific heat. The interatomic potential developed here can be used with some confidence to study physical properties of forsterite as a function of pressure and temperature.     

Rectangular parallellepiped resonance method for piezoelectric crystals and elastic constants of alpha-quartz

A theory is presented on the free osculations of a rectangular parallelepiped of piezoelectric crystal, by extending the theory of the rectangular parallelepiped resonance (RPR) method to determine elastic constants of crystals, as exemplified by an alpha-quartz specimen. The piezoelectric contribution to resonance frequencies was examined numerically on the specimen, and it was revealed that piezoelectricity causes approximately 5 kHz increase around 1 MHz. The resonance frequencies of the specimen were measured and inverted to elastic constants by least squares inversion. The inversion was by both the previous non-piezoelectric or elastic theory and by the present piezoelectric theory. The use of the non-piezoelectric theory resulted in an overestimate of 2σ or 0.6% in c ₁₁ and underestimate of σ or 6% in c ₁₂. These are the constants expected to be most affected by piezoelectricity. Errors are less than σ in the other constants. During measurement, it was found that the force applied to hold the specimen caused deviations from free oscillation and experimental errors of 5 kHz. The correction for this force is of some importance in RPR studies of piezoelectric crystals.     

Lattice dynamics of pyrite FeS2 — polarizable-ion model

Lattice dynamical calculations of the pyrite FeS₂ were performed using the polarizable-ion model (PIM) with different sets of short-range force constants. Not until the mean deviations between the observed and the calculated phonon energies become smaller than 3 cm^-1, the true force field can be established. In the case of only slightly greater deviations, the force fields computed differ strongly being without any physical meaning. The results are discussed with respect to the force constants K _i , F _i , and H _i , the effective dynamic charges and polarizabilities of the atoms involved, and the eigenvectors and potential energy distributions of the phonon modes. The most important short-range force constants are K ₁ (Fe-S stretching): 0.5 N cm^-1, K ₂ (internal stretching of the S₂ units): 1.0 N cm^-1, F ₁ (Fe^....Fe stretching): 0.2 N cm^-1, which indicate repulsive interactions of Fe atoms due to the occupied t _2g orbitals despite the relatively large Fe?Fe distances of 383 pm, and F ₂ and F ₃ (both intermolecular S₂?S₂ interactions): 0.2 N cm^-1. The great TO/LO splittings of some of the IR allowed phonon modes (species F _u) are caused by the large polarizabilities (2.4^.10⁶ and 3.3^.10⁶ pm³) of the atoms involved rather than by their effective charges (Fe: 0.2 e).     

Lattice dynamics of α-quartz including the effect of the width of the atomic electron distribution

On the basis of the polarizable-ion model (PIM) the effect of the width of the atomic electron distribution on the lattice dynamics of α-quartz was taken into account using the Birman method. The frequencies of optical mode, elastic constants and piezoelectric constants were reproduced using the parameters of this model (an effective charge z, a width parameter w, an electronic polarizability α, and force constants). The width parameter of oxygen atom was found to be about 0.59 Å, and the effective charge of the silicon atom 1.42 electron charges.     

A potential model for fluorapatite

For the mineral fluorapatite, Ca₁₀(PO₄)₆F₂, a potential model of the polarizable ion type has been developed in order to reproduce the following physical constants: the vibrational frequencies (transverse and longitudinal), the elastic constants and the static and high frequency dielectric constants. The parameters of the model are short range force constants, ionic charges and ionic polarizabilities. The polarizable ion model was built up in three stages in order to facilitate comparison with simpler models. In the first stage a short range valence type model was set up, assuming interaction up to a distance of 3.5 Å. In the second stage this model was extended to a rigid ion model by the addition of long range Coulomb forces. In the third stage, which resulted in the polarizable ion model, the polarizabilities of the ions were also taken into account. There was good agreement between calculated and observed data. This demonstrates that even for a crystal structure as complicated as that of fluorapatite with 42 atoms per unit cell a potential model can be constructed which reproduces in a satisfactory way the above-mentioned physical constants.     

Computer modelling of Al/Si ordering in sillimanite

Computer simulation is used to investigate the effect of Al/Si disordering over the tetrahedral sites on the lattice energy and the lattice constants of the mineral sillimanite Al₂SiO₅. A methodology for an atomistic assessment of the energy of the reaction 2(Si-O-Al)→(Si-O-Si)+(Al-O-Al) and its various contributions is established. This ordering energy is 0.97 eV for nearest neighbour sites in the ab-plane and 0.56 eV for those separated in the c-direction. The large difference is due to a greater constraint on the atomic relaxation in the ab-plane and shows the structural dependence of the ordering energy. Its magnitude appears to be determined by a complicated balance between Coulomb and short-range repulsive energy involving strain over many bonds, both in the ordered and disordered structures. There is also a significant interaction between second neighbour sites whereas the contribution of more distant neighbours is negligible. The lattice energies of most of the 154 configurations studied show a linear behaviour as a function of short-range order, specified by the number of Al-Al pairs. The ordering temperature T_c, estimated on the basis of a statistical mechanical model of disordering, and the calculated ordering energies are in semi-quantitative agreement with experimental values.     

Phononspectroscopy and lattice dynamical calculations of anhydrite and gypsum

Detailed experimental and theoretical studies of the k=0 vibrational spectra of anhydrite and gypsum are reported. The dielectric constants and the infrared reflection and Raman spectra of the single crystal have been measured. The frequencies of the phonon spectrum, the contribution of potential terms to potential energy and detailed mode assignments have been determined based on a polarizable-ion model. The relative experimental intensity of the spectra, the observed crystal field effects, the rotatory lattice modes of the H₂O molecule and the mixed character of translatory and rotatory modes of the Ca, SO₄, and H₂O groups are discussed based on the theoretical vibrational modes and the potential energy distribution. The principal moments of inertia of the water molecule and the Lennard-Jones potential constants for nonbonded oxygen-oxygen interactions are presented.     

Theoretical Analyses of the Infrared Spectrum of Baiyuneboite—（Ce）

Baiyuneboite-(Ce) is a new fluor-carbonate mineral.Based on the fine data on the structure of the mineral,the factor group and normal coordinate analyses for its infrared spectrum have been carried out.The factor group analyses indicate that the site group and factor group splittings of the internal vibration bands of CO3^2- ions do not occur and that the double bands of normal modes result from two non-equal sets of CO3^2- ions in the crystal structure,The normal coordinate analyses give the stretching force constants.bend force constants and calculated frequencies of CO3^2- ions.The calculations of potential energy distribution allow us to assign v3 and v4 to the stretching vibration and the bend vibration of CO3^2- ions.respectively.     

Variation of the elastic constants of tourmaline with chemical composition

Elastic wave velocities and lattice parameters of five tourmaline specimens with different chemical compositions have been measured. The piezoelectric effects on the elastic constants have been found to be small and can be neglected. Variations of the elastic wave velocities and elastic constants of the different tourmaline specimens indicate that: (i) partial substitution of Al by Fe in the structure decreases the shear wave velocities, (ii) replacement of Na by Ca increases the resistance of the structure against shear deformation involving C ₆₆, (iii) replacement of Al by Mg seems to decrease the resistance of the structure against longitudinal deformation involving C ₃₃. Elastic constants C ₁₁, C ₃₃, C ₄₄ and C ₆₆ of the different tourmaline specimens used in this study differ individually by 1.7 percent to 6.7 percent, indicating that the large differences (up to 21%) between the values reported by previous authors cannot be explained in terms of the chemical composition alone.     

Computer simulations of the structural and physical properties of the olivine and spinel polymorphs of Mg2SiO4

The aim of the work presented is to develop a computer simulation technique which will predict the structure and physical properties of forsterite and ringwoodite, the major mantle-forming polymorphs of Mg₂SiO₄. The technique is based upon energy minimization, in which all structural parameters are varied until the configuration with the lowest energy is achieved. The lattice energy and physical properties (e.g. elasticity and dielectric constants) are calculated from interatomic potentials, which generally include electrostatic and short-range terms. We investigate several types of traditional potential models, and present a new type of model which includes partial ionic charges and a Morse potential to describe the effect of covalency on the Si-O bond. This new form of potential model is highly successful, and not only reproduces the zero-pressure structural, elastic and dielectric properties of forsterite and ringwoodite, but also accurately describes their pressure dependence.     

The extrapolation of elastic moduli to high pressure and temperature

The elastic constants of a crystal under stress, defined as the second derivative of the crystal free energy with respect to strain, require a correction related to the static pressure at non-zero pressures. The corrections required for the elastic constants calculated by the free energy minimisation code PARAPOCS are described and tested by comparison with the elastic constants calculated numerically by applying small stresses in the appropriate orientations to simulated crystals of fluorite, forsterite, α-quartz and albite. The corrected elastic constants are then used to investigate the extrapolation of the bulk and shear moduli (and hence also the seismic wave velocities V _p and V _s) of β-spinel and forsterite to upper mantle pressures. A Murnaghan equation, thirdorder Eulerian finite strain equation, second order polynomial equation and a logistic equation were all fitted to the simulated bulk and shear moduli between 0 and 3 GPa pressure. The parameters derived for these equations are used to extrapolate the bulk and shear moduli to 14 GPa and the results are compared to the simulated high pressure moduli. Over this pressure range, the second order polynomial provides the best extrapolation of the bulk modulus, but the use of the logistic equation results in the best extrapolation of the shear modulus.     

Molecular dynamics study of the α–β transition in quartz: elastic properties, finite size effects, and hysteresis in the local structure

The α–β transition in quartz is investigated by molecular dynamics simulations in the constant stress ensemble. Based on a frequently used two-body interaction potential for silica, it is found that anomalies in the elastic constants are at least in semiquantitative agreement with experiment despite the fact that no anomaly in the c/a ratio is observed in the simulations. A finite-size scaling analysis shows that first-order Landau theory is applicable to the employed model potential surface. This statement also applies to the susceptibility below the transition temperature T _tr, which has not yet been measured experimentally. Examination of the local order near T _tr reveals that the deformation of SiO₄ tetrahedral units is equally large in the β phase as in the α phase. However, large hysteresis effects can be observed in the local structure for distances r > 4 Å. The results are in agreement with the picture of a first-order displacive phase transformation which is driven by the motion of deformed tetrahedral SiO₄ units. Yet, the fast oscillations of oxygen atoms are around (time-dependent) positions that do not correspond to the ideal oxygen positions in β-quartz. The averaged configurations resemble the ideal structure only if averaged over at least a few nanoseconds.     

Electric field gradient calculation in forsterite,Mg2SiO4

The electric field gradient (EFG) in Mg₂SiO₄ is calculated on the basis of the extended point ion model, including the local term of the overlap contribution. The agreement with experimental data deduced from the quadrupole coupling constants and principal axes at the Mg sites is quite good. The results of the present calculation exhibit a small overlap contribution to the EFG at M1 and a clearly bigger one at M2, whereas the lattice contribution to the EFG at M1 and M2 is reversed. The distinct overlap effects are assumed to be due to the particular Mg₂SiO₄ crystal structure and the different point symmetry at M1 and M2. The oxygen polarizability and charge used to calculate the EFG tensor were found to be smaller than the theoretical polarizability and formal charge, respectively. The sign of the Mg quadrupole coupling constants at M1 and M2, which has not been determined experimentally, results from the EFG calculation as positive.     

Calculation of the equation of state and elastic moduli of MgO using molecular orbital theory

The success of molecular orbital theory in calculating crystal properties such as bond lengths and atomic force constants has been well documented in the literature. Calculations can be extended to crystals under simulated compression and strain to determine elastic moduli and their pressure derivatives. Comparison of the molecular orbital results with both experimental values and results obtained by calculations such as the potential induced breathing model provides insight into the nature of chemical bonding in MgO. In this study, several molecular clusters were investigated as possible models for MgO; the cluster Mg₄O₄H₂₄ was chosen as the best model. Molecular energies were calculated with respect to bond length for both compression and expansion based on clusters that had been optimized for minimum energy. The resulting energy-volume curve was fitted to a recently derived equation of state (Brown, in preparation) to derive the values of K ₀ and dK₀/dP and the individual elastic moduli and their pressure derivatives were calculated by applying strain to the molecular cluster at both zero and elevated pressures. Agreement between theory and experiment varies between parameters, but the overall trend is encouraging. Since the molecular orbital model includes only short range interactions, its ability to approximating model the elastic moduli of MgO suggests a strong contribution to the elastic energy from short range interactions.     

Interatomic potentials for CaCO3 polymorphs (calcite and aragonite), fitted to elastic and vibrational data

Calcite and aragonite have been modeled using rigid-ion, two-body Born-type potentials, supplemented by O-C-O angular terms inside the CO₃ groups. A shell model has also been developed for calcite. Atomic charges, repulsive parameters and force constants have been optimized to reproduce the equilibrium crystal structures, the elastic constants and the Raman and infrared vibrational frequencies. The rigid-ion potential RIM (atomic charges:z _O= -0.995e,z _C = 0.985e,z _Ca = 2.0e) fitted to calcite properties is able to account for those of aragonite as well. Experimental unit-cell edges, elastic constants, internal and lattice frequencies are reproduced with average relative errors of 2.1, 5.5, 2.4, 15.1% for calcite and of 0.2, 19.4, 2.5, 11.8% for aragonite, respectively. The RIM potential is suitable for thermodynamic and phase diagram simulations in the CaCO₃ system, and is discussed and compared to other potentials.     

Elasticity of single-crystal MgAl2O4 spinel up to 1273 K by Brillouin spectroscopy

The adiabatic single-crystal elastic constants, C _ij , of stoichiometric magnesium aluminate spinel (MgAl₂O₄) have been measured up to 1273 K by highresolution Brillouin spectroscopy, using a 6-pass tandem Fabry-Pérot interferometer and an argon ion laser (514.5 nm). Two platelet samples were employed for probing the acoustic phonons along [100] and [110] directions by platelet and backscattering geometries. The measured temperature dependences of the elastic moduli show a distinct anomaly at 923 K in the shear modulus C _s = (C₁₁-C₁₂)/2 (along [110] direction) and the longitudinal modulus C₁₁ (along [100] direction). This anomaly is consistent with the order-disorder phase transition, resulting from the atomic exchange between Mg at the tetrahedral site and Al at the octahedral site, which has been well documented recently (Peterson et al. 1991; Millard et al. 1992) by neutron powder diffraction and ²⁷Al magic-angle spinning NMR. The values of the temperature derivatives of v _p , v _s , and K _s , in the temperature range 300–923 K, calculated by the Voigt-Reuss-Hill approximation are -0.40ms^?1 K^?1, -0.26ms^?1 K^?1, and -1.89 x 10^?2GPaK^?1.     

Molecular Orbital Study on the Optimized Geometries and Spectroscopic Parameters of Borate Polyhedra

In this paper several methods including MNDO, multiple scattering Xα and ab initio self-consistent-field MO theories have been used to calculate the minimum energy geometries, force constants, vibrational frequencies ,and ^11B quadruple coupling constants of B-O polyhedra such as [BO3],[BO4],[OB2] and [OB3].The results are in good agreement with the experimental and calcu-lated values so far published by other authors.     

Elastic and thermoelastic properties of synthetic Ca<Subscript>2</Subscript>MgSi<Subscript>2</Subscript>O<Subscript>7</Subscript> (åkermanite) and Ca<Subscript>2</Subscript>ZnSi<Subscript>2</Subscript>O<Subscript>7</Subscript> (hardystonite)

Elastic and thermoelastic constants of large single crystals of Ca₂MgSi₂O₇ and Ca₂ZnSi₂O₇ have been derived from ultrasonic resonance frequencies of plane-parallel plates and their shift upon variation of temperature, respectively. In addition, coefficients of thermal expansion and dielectric constants were determined. Both species possess quite similar properties. As observed in other isotypic magnesium and zinc compounds, the mean elastic stiffness and the deviation from the Cauchy relations are significantly larger in the zinc compound, due to a covalent contribution of the Zn–O bond. Positive thermoelastic constants T₄₄ and T₆₆ in Ca₂MgSi₂O₇ allow temperature-independent ultrasonic generators and oscillators to be manufactured.